Philippe Guicheteau "Bifurcation theory: a tool for nonlinear flight dynamics", Phil.Trans.R.Soc.Lond.A (1998) 356, 2181-2201 This paper presents a survey of some applications of bifurcation theory in flight dynamics at ONERA (France). After describing basic nonlinear phenomena due to aerodynamics and gyroscopic torque, the theory is applied to a real combat aircraft, and its validation in flight tests is shown. Then, nonlinear problems connected with the introduction of control laws to stabilize unstable dynamic systems and transient motions are addressed.

1. Bifurcation theory: a tool for
nonlinear flight dynamics
By Philippe Guicheteau
ONERA, B.P. 72, 92322 Chatillon CEDEX, France
This paper presents a survey of some applications of bifurcation theory in flight
dynamics at ONERA. After describing basic nonlinear phenomena due to aerody-
namics and gyroscopic torque, the theory is applied to a real combat aircraft, and its
validation in flight tests is shown. Then, nonlinear problems connected with the intro-
duction of control laws to stabilize unstable dynamic systems and transient motions
are addressed. To extend the scope of applications, ongoing research devoted to the
analysis of complex dynamic systems, including both continuous and discrete time
parts, is mentioned. In conclusion, as a result of work undertaken at ONERA, it is
stated that this theory is a useful tool for the study and control of high-dimensional
dynamic systems.
Keywords: bifurcation theory; combat aircraft; dynamic systems;
flight dynamics; flight tests; stability analysis
1. Introduction
It is now well known that the bifurcation theory can help predict the asymptotic
behaviour of nonlinear differential equations depending on parameters. Efficient
numerical procedures are now available and several previous studies have demon-
strated that bifurcation analysis can be used to predict complex phenomena.
As in flight dynamics, aircraft motion is described by a set of nonlinear differential
equations, depending on parameters, associating the state vector (angle of attack
(AOA), sideslip angle, speed, angular rates, etc.) with the control vector (motivators,
etc.) through motion equations, aerodynamic models, and flight-control systems. This
paper aims to present results obtained in this field with a global stability analysis
methodology making use of bifurcation theory.
After a brief presentation of the methodology and numerical procedures available,
basic but very simple and well-known nonlinear phenomena, such as spiral mode,
auto-rotational rolling, and Dutch-roll instability, are revisited by using bifurcation
theory.
Then, the theory is applied to a real combat aircraft, the German–French Alpha-
Jet from Dassault Aviation. After a brief description of the aircraft model, the oscil-
latory flight cases, such as ‘agitated’ spins, are studied by means of learning the
stability characteristics of periodic orbits related to oscillatory unstable equilibrium
points. Complex oscillatory modes are pointed out. The synthesis of all this illus-
trates that the lack of a realistic nonlinear model may lead to great difficulties for
flight analysis when the motion is quasi-periodic or chaotic. Comparisons between
predictions and flight tests at the French flight test centre are shown.
Control laws either stabilize unstable systems and/or increase their robustness
under system modifications and perturbations. In practical situations, nonlinearities
Phil. Trans. R. Soc. Lond. A (1998) 356, 2181–2201
Printed in Great Britain 2181
c 1998 The Royal Society
TEX Paper

2. 2182 P. Guicheteau
are numerous either in the dynamic system or in its control laws. Rather than detail-
ing nonlinear control theory, it is shown that it can be very instructive to introduce
bifurcation analysis while designing control, bearing in mind that the robustness of
a stable steady state is closely related to its region of asymptotic stability.
From a theoretical point of view, parameter variations are assumed to be fixed
and independent of time. When temporal parameter variations are not small, one
can observe behaviour that is different from that initially predicted by means of
bifurcation theory. In § 6 we discuss the connection between asymptotic behaviour
and quasi-stationary and/or transient behaviour. These considerations are closely
connected with the attracting-basin computation problem, which is also addressed.
The previous results have been obtained with continuous or ‘almost-continuous’
dynamic systems. To extend the scope of application in a more realistic way, it is
necessary to be able to work with discrete-time systems and, finally, with complex
dynamic systems including both a continuous part and a discrete-time part. Ongoing
research devoted to the analysis of complex dynamic systems at ONERA is proposed
in § 7.
2. Methodology
During the past 10–15 years, many methods have been suggested for the numerical
solution of nonlinear problems (Guicheteau 1993a). This includes, in particular, the
solution of parameter-dependent nonlinear equations by continuation techniques, and
the related methods for bifurcation and stability analysis. Some of these codes deal
with several aspects of the problem, while others concentrate only on specific aspects.
All these codes are based on powerful continuation methods which are a direct result
of the implicit-function theorem. However, it is noticed that continuation requires
evaluation of the system and computation of partial derivatives, which can be very
time consuming. Thus there is a need to improve the performance of such codes in
order to get almost interactive procedures, even for high-dimensional systems. To this
end, and for several years, an efficient numerical code, ASDOBI, for the analysis of
continuous and discrete-time nonlinear-dynamic systems using robust continuation
algorithms has been developed at ONERA (Guicheteau 1993a).
Once a numerical procedure is chosen, one has to set up a methodology to inves-
tigate the behaviour of the dynamic system.
The first step is to compute all the steady solutions of the system and their asso-
ciated stability. As this step is generally time consuming, the computations have to
be limited to the field of interest. Nevertheless, one must be very careful because the
number of steady solutions for a given parameter is generally not known. Therefore,
a priori qualitative experience on the global behaviour of the systems is preferable.
The second step consists of making graphic representations of the results in appro-
priate subspaces especially for high-dimensional systems. This step requires versatile
graphic codes, and, again, a good experience of the system under consideration.
Sometimes this step shows that equilibrium branches are missing.
The third step is concerned with the prediction of system behaviour when a bifurca-
tion point is encountered. To achieve this, the computation of the attracting domain
of the stable steady-states and, once more, the experience of the engineer, are very
useful when investigating the various possibilities.
Phil. Trans. R. Soc. Lond. A (1998)

3. Bifurcation theory 2183
Experience acquired by the processing of a large number of cases is extremely valu-
able for correctly predicting system behaviour. Thanks to this experience, it is not
always necessary to check the predictions by means of numerical simulations. How-
ever, for the difficult cases, and if there is any doubt, the last step of the methodology
consists of performing only a few numerical simulations before testing the predictions
in simulation and on the real system.
3. Basic nonlinear phenomena
The prediction and analysis of control loss and spin of aircraft are old problems.
However, with the lack of computer capabilities, previous techniques involved the
application of exact or approximate analytical methods to simplified nonlinear equa-
tions, taking into account gyroscopic torque and some aerodynamic nonlinearities
(Phillips 1948; Pinsker 1958; Hacker & Oprisiu 1974; Kalviste & Eller 1989; Ross &
Beecham 1971; Padfield 1979; Adams 1978; Laburthe 1975).
The studies of Schy & Hannah (1977) and Schy et al. (1980) can be considered
as two of the last works that can be related to a simplified treatment of control
losses. They showed the possibility of multiple equilibrium solutions for a simplified
set of flight equations, several of which are stable. These works formed the basis
of the global methodology used at ONERA. These results have been revisited and
completed by many scientists in the past two decades (see Guicheteau (1993a) and
Littleboy & Smith (1997) and references therein). Nevertheless, most of the results
presented in previous papers are similar, and exhibit basic nonlinear phenomena that
are useful in gaining a better understanding of aircraft behaviour.
(a) Spiral instability
This slow motion occurs at low AOA when the aerodynamic model is symmetrical.
Only gravity and pitch angle have an effect on the stability of the motion at zero
sideslip angle.
When lateral control deflections are at neutral, the equilibrium surface of the
system versus elevator deflection shows that spiral instability is bounded by two fork
bifurcations on the lateral variables of the system (Guicheteau 1993a). Between these
two limits of stability, the aircraft is unstable in straight level flight and, in response
to a lateral disturbance, it tends towards a turning-down flight, the characteristics
of which are determined by the stable equilibrium branches located between the two
limits of stability at zero sideslip angle (figure 1).
By comparison with the classical linearized flight dynamics, this methodology is
able to predict system behaviour beyond the limit of stability. It also gives an idea
of the nonlinearities responsible for the instability. Thus, considering the essential
nonlinearities, it is possible to analyse aircraft behaviour by reducing the motion
equations to a scalar equation relating roll angle, pitch angle, and control deflections:
˙Φ = (A sin Θ + B cos Θ cos Φ) sin Φ + (Cδaδa + Cδrδr) cos Φ + Dδaδa + Dδrδr,
in which A, B, C and D are coefficients that depend on the aerodynamic charac-
teristics. In particular, B is the classical stability criterion for the linearized motion
equations.
Despite its simplicity, this formulation synthesizes aircraft behaviour in the vicinity
of the spiral instability. Moreover, it shows that spiral bifurcation can appear in the
aileron–rudder plane even if level flight at zero sideslip angle is stable (figure 2).
Phil. Trans. R. Soc. Lond. A (1998)

4. 2184 P. Guicheteau
Figure 1. Spiral bifurcation: ——, stable; – – –, unstable divergent.
Figure 2. Spiral bifurcation in aileron–rudder plane.
(b) Auto-rotational rolling
Experiments and previous computations have shown that auto-rotational rolling
occurs at low AOA. It can also be seen that speed varies only a little and that the
influence of gravity is negligible. Then, assuming also that pitch rate and yaw rate
are much smaller than the roll rate, it is still possible to transform the analysis of the
motion equations into the study of a polynomial nonlinear scalar equation, similar to
the canonical form of a singularity, which is called a ‘butterfly catastrophe’, relating
yaw rate and control deflections:
˙p = f6(δe)p5
+ (f5δa
δa + f5δr
δr)p4
+ f4(δe)p3
+ (f3δa
δa + f3δr
δr)p2
+ f2(δe)p + (f1δa
δa + f1δr
δr),
in which the coefficients fi depend on aerodynamic characteristics and gyroscopic
torques (figures 3 and 4).
Taking into account gravity and speed effects, the real behaviour of the aircraft
shows that gravity effects result in an oscillation of the state variables around a mean
value, and cause the aircraft to dive and accelerate. It follows that the roll rate does
not stabilize at the predicted value. More precisely, it is observed that it is possible
to rewrite the system under study by using the reduced roll rate and verify that,
despite the increase in aircraft speed and roll rate, the value of the reduced roll rate
is perfectly stabilized at the value anticipated by the equilibrium computations. The
reduced angular rate is therefore a good indicator of the behaviour.
Phil. Trans. R. Soc. Lond. A (1998)

5. Bifurcation theory 2185
Figure 3. Inertia coupling: steady roll rate versus aileron deflections:
——, stable; – – –, unstable divergent.
Figure 4. Inertia coupling: bifurcation surface in aileron–elevator plane.
Moreover, once reduced to a scalar equation, a control system which avoids bifurca-
tions can be easily designed. In practice, such a system is an aileron–rudder coupling,
which is used on many aircraft. Although it does not modify the pattern of the bifur-
cation curves, that is intrinsic to the aircraft, it does modify their occurrence; the
possible equilibria are no longer the same, and are not so varied.
(c) Dutch-roll instability
This phenomenon is very well reported in the literature (Guicheteau 1993a). In
this case, instability is connected to a Hopf bifurcation point, which is approximated
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6. 2186 P. Guicheteau
Figure 5. Equilibria of a submarine model. ——, Stable; – – –, unstable divergent;
–·–·–, oscillatory unstable.
by classical theoretical and experimental handling quality criteria (cnβdyn, Kalviste,
etc.).
In this case, the first point of interest of bifurcation theory is the characterization
of the Hopf bifurcation (subcritical or supercritical) (Guicheteau 1986) in order to
get an indication of the evolution of the amplitude of the periodic motion beyond the
limit of stability. Then it is of interest to compute the periodic-orbits envelope, with-
out the usual simplified assumptions, in order to investigate secondary bifurcations
which can take the appearance of wing rock or spin.
(d) More complex phenomena
So far, it has been shown that bifurcation theory is a powerful tool for understand-
ing several classical nonlinear flight-dynamics phenomena, for which a linearized
approach is not suitable. Similar phenomena have been also exhibited at ONERA
with models of submarines for which the aerodynamic part is replaced by an hydro-
dynamic one. As an illustration, figure 5 presents the computed equilibria of a sub-
marine model during an emergency manoeuvre with degraded control (Pavaut 1993).
Several of them are very similar to the inertia coupling and spin of aircraft. However,
as on combat aircraft, more complex phenomena are also encountered.
Computation of periodic-orbit envelopes and their bifurcations shows that much
agitated behaviour can also be analysed by means of bifurcation theory. In particular,
Phil. Trans. R. Soc. Lond. A (1998)

7. Bifurcation theory 2187
when two conjugate imaginary eigenvalues of the transition matrix of a periodic
orbit cross the unit circle while a parameter varies, the initial stable orbit becomes
unstable and the motion lies on a toroidal surface surrounding the newly unstable
orbit. Generally, in this particular case, the motion seems to be a superposition of
two periodic motions with very different periods. Finally, when all the equilibrium
states are unstable except for one which is weakly stable, very long transient motion
can appear. Sometimes, this motion seems to be complex and/or chaotic. So this
could result in some difficulties when analysing and modelling such phenomena from
flight tests because the running time is generally less than the larger period. More
precisely, and in case of sensitivity to initial conditions, if the identification process
ignores the existence of such motion, and the influence of typical nonlinearities, it
then results in an identified model that matches only the flight-test data used in the
identification process. It will not exhibit the sensitivity to initial conditions.
4. Application to a real combat aircraft
The Alpha-Jet is a tandem two-seat German–French aircraft for close support and
battlefield reconnaissance. With narrow strake on each side of the nose, it is also an
advanced jet trainer. Considering its great ability to safely demonstrate numerous
and various high-AOA behaviours and for flight-test correlation, the training version
was chosen to investigate the interest of the methodology (Guicheteau 1993b).
(a) Aircraft model
Each of six global aerodynamic coefficients is expressed independently as a function
of flight and control parameters. A general expression of coefficients can be expressed
in the form,
ci = ci stat + ci unstat,
where the coefficients ci stat and ci stat represent stationary aerodynamic effects and
take into account unsteady effects expressed as a transfer function, respectively. All
these terms have been measured and tabulated over a wide state and control domain.
Generally, in order to prevent continuation problems, aerodynamic coefficients are
usually smoothed to ensure continuity and derivability conditions for the resulting
nonlinear dynamic system. In our application, no preliminary smoothing was done
to avoid pure numerical bifurcation points and behaviour. Coefficients are evaluated
by linear interpolation of the tabulated data.
(b) Predictions and flight tests
The results presented here are related to control losses, spin and spin recovery.
In order to simplify the interpretation of computations, only typical cases will be
shown. Before discussing the results, it should be noticed that they are extracted
from classified studies. So the angular rates are plotted without a scale.
Starting from a straight level flight at low AOA, when the pilot moves the elevator
for a full nose-up attitude, we can observe multiple steady-states appearing at high
AOA when both aileron and rudder deflection vary. The projection of the equilibrium
surface in characteristic subspaces easily allows identification of the domains of spin
Phil. Trans. R. Soc. Lond. A (1998)

8. 2188 P. Guicheteau
Figure 6. Equilibrium surface projection in a characteristic subspace
(aileron, rudder, incidence).
and rolling motions (figure 6). It should be observed that this surface is not sym-
metrical. This is due to non-symmetrical aerodynamic data for symmetrical aileron
and rudder deflections at high AOA.
As can be seen in figure 6, stability is very different from one point to another.
More precisely, it seems that left spins, related to negative aileron deflections, are
much more unstable than right spins. Perhaps this low degree of stability can explain
pilot difficulties in demonstrating left steady spins on an Alpha-Jet. The third type
of steady-state encountered on this surface, corresponds to an important roll motion
at moderate AOA. In flight, this kind of motion occurs mainly when pilots fail spin
entry or fail the transition from one spin on one side to another spin on the other
side.
Considering the equilibrium curve for full rudder deflection (figure 7), it can be
seen that right spin is stable while left spin is always oscillatory unstable except for
a few positive aileron deflections.
Surrounding this last equilibrium branch for negative aileron deflections, there
exist several periodic orbits when aileron deflection decreases. The amplitude of
AOA and roll rate of the computed orbits are plotted versus aileron deflections in
figure 8. Two distinct branches can be observed. On the first one, the limit points are
numerous. For small deflections, two convergent series of flip periodic bifurcations
determine a region in which an Alpha-Jet can exhibit chaotic behaviour. In our case,
contrary to typical chaotic behaviour exhibited by well-known particular differential
equations, there are only small differences in amplitude between the different orbits
of period T, 2T, etc. It seems then, that this behaviour will be very difficult to observe
and to characterize in flight. Finally, the most important phenomena on this branch
Phil. Trans. R. Soc. Lond. A (1998)

9. Bifurcation theory 2189
Figure 7. Equilibrium curve for δr = 17◦
: ——, stable; · · · · · · , unstable divergent;
–·–·–, oscillatory unstable.
of the envelope is the rapid variation of orbit amplitude with aileron deflection. This
could explain the level of agitation, which is well known to pilots.
On another branch of the periodic envelope, and for deflections from −10 to −20◦
,
there exist oscillatory unstable orbits with great amplitude (figure 8). Around them,
the motion takes place on a toroidal surface if it is stable. Nevertheless, the existence,
for some deflection of an invariant torus and a stable orbit, can lead to non-similar
flight behaviour. This different behaviour depends on the initial state and on the
history of control deflections during the manoeuvre.
All these phenomena have been demonstrated in simulation. In order to make a
correlation between predictions and flight, flight tests have been done at the French
Flight Test Centre in Istres. Before presenting a few of the results obtained during
the flight tests, one must bear in mind that good correlation indicates only that the
aircraft model is realistic. It is not a validation of the methodology which had been
already validated through numerical simulations.
For full nose-up elevator and positive rudder deflections, quiet left spin is obtained
for a small aileron deflection (figure 9). When aileron deflection is close to −10◦
, the
Alpha-Jet can exhibit three very different motions due to the existence of two stable
orbits and a stable invariant torus (figures 10 to 11).
Finally, it can be noticed that chaotic motion for small aileron deflections was
not demonstrated. This result is due to the short duration of spin tests and to the
absence of great differences between the orbits in presence, as previously mentioned.
Nevertheless, this phenomenon was known by the pilots who experienced it, and who
knew that it was impossible to achieve similar flight tests with such deflections.
Phil. Trans. R. Soc. Lond. A (1998)

10. 2190 P. Guicheteau
Figure 8. Amplitude of periodic orbits when δa varies between −20 and −17◦
:
——, stable; · · · · · · , unstable divergent; –·–·–, oscillatory unstable.
5. Some remarks about control problems
As opposed to many theoretical approaches, practical control laws are generally
nonlinear because of their formulation or the use of nonlinear elements. Thus, it is
interesting to investigate whether the proposed methodology can help the designer
to design ‘good’ control laws from a stability point of view. More precisely, one must
answer the following question: can ‘stabilizing’ control laws introduce new bifurca-
tions while they are used to ‘stabilize’ the open-loop system?
Without invoking bifurcation theory, these control problems are being studied and,
fortunately, ‘good’ solutions have been found in many particular cases. A considerable
number of results are already available in the literature.
In this section we are concerned with an autonomous dynamic system:
˙x(t) = f(x(t), u(t)),
where x and u denote state and control vectors, respectively. f and x are n-di-
mensional vectors, u is an m-dimensional vector and f(x(t), u(t)) are nonlinear
functions satisfying Lipschitz conditions in which the control vector depends on the
Phil. Trans. R. Soc. Lond. A (1998)

11. Bifurcation theory 2191
Figure 9. Quiet left spin for δa = −4◦
.
state variables in the following way:
u(t) = g(x(t), p),
where p represents new control parameters and g(x(t), p) are nonlinear functions
satisfying Lipschitz conditions. From the definition of the control vector, it follows
that every equilibrium point of the open-loop system is also an equilibrium point
of the closed-loop system and vice versa. However, the stability of the closed-loop
solutions can differ from the stability of the open-loop solutions and new complex
asymptotic solutions may appear.
As a first example, let an unstable second-order linear differential equation be
stabilized by a control law which depends on a nonlinear element as described in
figure 12.
Phil. Trans. R. Soc. Lond. A (1998)

12. 2192 P. Guicheteau
Figure 10. Regular agitated spin for δa = −10◦
.
A linearized analysis of the closed-loop system,
¨x + 2ζ0ω0 +
∂k( ˙x)
∂ ˙x
˙x + ω2
0x = e,
shows how gain influences the stability of the asymptotic solution. When (∂k( ˙x)/∂ ˙x)
goes from zero to the adopted value, k∗
, for the closed-loop system, it crosses a value
kc
for which the steady-state admits two conjugate pure imaginary eigenvalues.
From a nonlinear point of view, one can say that there is a Hopf bifurcation in kc
.
Moreover, consider a nonlinear gain,
k( ˙x) =
2
π
k( ˙x)lim arctan
kπ ˙x
2k( ˙x)lim
,
i.e. the feedback is almost linear (k( ˙x) ∼= k) in the vicinity of the equilibrium points
and is saturated (k( ˙x) ∼= k( ˙x)lim) when ˙x tends to infinity. Then, one can easily show
that the Hopf bifurcation is subcritical. Thus, when k is greater than kc
, periodic
orbits surround the stabilized equilibrium points. These orbits can exist even for the
adopted gain value k∗
(figure 13).
It can be seen, in this two-dimensional case, that the unstable periodic-orbit enve-
lope visualizes the boundary of the attracting region of the controlled system. As
an example, if the amplitude of a perturbation is greater then the amplitude of the
periodic orbit, the controlled system exhibits a divergence.
Applied to a typical combat aircraft with unstable lateral modes that is stabi-
lized by a continuous feedback with a saturation on angular rates and stops on
Phil. Trans. R. Soc. Lond. A (1998)

13. Bifurcation theory 2193
Figure 11. Motion on a toroidal surface for δa = −10◦
.
Figure 12. Controlled unstable second-order dynamic system.
lateral control deflections, computation of the unstable orbit surrounding the stabi-
lized equilibrium point gives a first insight into the ‘robustness’ of the control law
(figure 14). Moreover, stops on control deflections generate a stable periodic orbit
which surrounds the unstable limit cycle and limits the divergence of the motion. As
the dimension of the system is greater than two, the unstable periodic orbit is only
one element of the attracting-region boundary.
Similar classified results had been obtained at ONERA when this methodology was
applied to a modern combat aircraft and to a realistic model of an air-to-air missile
for which the dimension of the system is greater than 50. Except for the difficulty
related to the dimension of the system, the true difficulty was related to the numerical
procedure because the real nonlinear elements were not C∞
functions (saturation,
Phil. Trans. R. Soc. Lond. A (1998)

14. 2194 P. Guicheteau
Figure 13. Amplitude of the steady solutions as a function of the nonlinear control gain:
——, stable; – – –, unstable; –·–·–, oscillatory unstable.
Figure 14. Fixed point and periodic orbits for a typical stabilized combat aircraft.
——, stable orbit; – – –, unstable orbit.
stop, hysterisis, etc.). So, it was necessary to improve the continuation algorithm
to work with such singularities. The first improvement consists of predicting a new
equilibrium solution form the previous one by using a nonlinear extrapolation based
on non-equidistant points. This makes the continuation process easier when the equi-
librium curve is highly nonlinear. The second major improvement is the ability of
the continuation process to detect when the system is independent of a variable or
a parameter, to automatically reduce the dimension of the system and to continue
the computation with the reduced system. This case is generally encountered when
controlled systems contain hysteresis effects and saturation on actuators. However, a
lot of work remains to be done in order to characterize the stability of such systems.
Now, many control laws are provided by a computer with a sampling period.
In order to increase the interest of the methodology for practical applications, it
is necessary to be able to investigate the behaviour of complex dynamic systems
which are composed of both a continuous part (motion equation) and a discrete time
part (control law). In particular, one needs to characterize the stability of perturbed
equilibria between sampling times. After solving the analysis of discrete-time systems,
Phil. Trans. R. Soc. Lond. A (1998)

15. Bifurcation theory 2195
ONERA studies are now investigating numerical procedures to analyse these complex
dynamic systems and new results will be available soon.
6. Attracting-basin and transient behaviour
A stable equilibrium state of a nonlinear-dynamic system is surrounded by a stability
region. The determination of this region is of great interest for dynamicists and
engineers. It allows definition of the limit of validity of linearized approximations for
the original nonlinear equations, better understanding of the global behaviour of the
system, and determination of the maximum values of the perturbations for which
the perturbed system returns to the initial stable state.
The aim of this section is to present several methods which are currently in use,
in order to give an answer to the attracting-basin computation problem for sets of
ordinary nonlinear differential equations.
(a) Preliminaries
The systems under consideration are of the general form:
˙x(t) = f(x(t)),
where f and x are n-dimensional vectors, and where f(x(t)) are nonlinear functions
satisfying Lipschitz conditions. It exhibits, at least, a steady state x∗
dx∗
(t)
dt
= 0, t > t0,
or a periodic orbit defined by
˙Φ∗
(t) = f(Φ∗
(t)),
where
Φ∗
(t + τ) = Φ∗
(t), τ = NT,
where N is a positive integer and T the period.
Considering a periodic orbit, and without loss of generality, we can assume that
x∗
= 0 and that t0 = 0 and notice that by defining ξ = x − Φ∗
(t), the study of
periodic solutions for a time-invariant system is transformed to the study of periodic
systems (Poincar´e map).
The domain of attraction (attracting basin, region of asymptotic stability) of a
steady-state or of a periodic orbit, is defined as the set of all initial conditions,
x0(t0), that tend, respectively, to x∗
or to Φ∗
(t) when time tends to infinity.
Numerous methods have been proposed in the literature for estimating the region
of asymptotic stability. They may be roughly divided into Liapounov and non-
Liapounov methods. Nevertheless, the distinction between these different classes of
methods is less obvious now because modern methods generally use joint approaches.
(b) Liapounov methods
The methods using Liapounov functions are derived from the results obtained by
Liapounov. Two approaches have been developed by using either results from Zubov
or an extension of Liapounov’s theorems produced by La Salle.
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16. 2196 P. Guicheteau
The first approach may be applied to low-dimensional nonlinear systems with
an exactly defined structure, and, generally, give conservative results. For higher-
dimensional systems, several optimization approaches are used to modify an initial
Liapounov function in order to enlarge the volume of the attracting region (Michel
et al. 1982).
The second approach is related to the application of the concept of absolute stabil-
ity in the frequency domain, as proposed by the Popov criterion, choosing a suitable
Liapounov function holding for a whole class of nonlinear systems defined by a sec-
tor condition in the sense of Aizerman. In this case, the results obtained with this
approach are specific to a type of nonlinear system but they are more general than
the previous one. Nevertheless, they are also too conservative.
(c) Trajectory-reversing method
This method is known as the trajectory-reversing method or backward mapping.
It is based on the La Salle extension of the Liapounov stability theory. It provides an
iterative procedure for obtaining the global attracting region for multi-dimensional
systems, both time-invariant and time varying without conditions on the topological
nature of the asymptotically stable point under study.
Once an initial estimation of the attracting region bounded by the curve c0 is made,
a curve Cj is obtained by backward integration in time of the dynamic equations
from t = 0 to t = tj. Then, if c−∞ denotes the map of the curve c0 as t → −∞, due to
the uniqueness of the solution of the system, c−∞ is the domain of attraction of the
origin. Generally, it is not necessary to compute c−∞ to get a good approximation
of the true domain of attraction.
The trajectory methods are attractive because of their generality and simple the-
oretical framework. However, their computational efficiency is generally poor and
they have only been used for low-dimensional systems.
To reduce the computational effort, (n − 1) facets can be used to approximate
the basin boundary of an nth order system (Piaski & Luh 1990). Starting from a
local quadratic Liapounov function around the stable equilibrium point under study,
a small convex polytope† is generated. Then the vertices of this initial polytope
are integrated backward in time to generate the vertices of a non-convex polytope
approximation of the basin boundary. Thus, the real image approximates the attract-
ing region as backwards integration time approaches infinity. However, as the system
is a nonlinear one, a test is applied to check whether the new non-convex polytope
is a good approximation of the image of the original convex polytope. Adaptative
facet refinement is used to correct any inaccuracy of the image approximation. Even
when applied to low-dimensional systems, a great number of vertices are still required
for accurate approximation. More generally, extension to higher-dimensional cases is
actually limited by the rapid growth of the (facet number)/(vertex number) ratio
with the growth of the state-space dimension.
(d) Differential geometry method
This method gives a complete characterization of the stability boundary for a fairly
large class of nonlinear autonomous dynamic systems satisfying two generic prop-
erties plus one additional condition that every trajectory on the stability boundary
† A polytope is a finite, flat-sided solid in any high-dimensional space.
Phil. Trans. R. Soc. Lond. A (1998)

17. Bifurcation theory 2197
Figure 15. Partial view of the boundary of the domain of stability.
approaches one of the equilibrium states (fixed points or/and periodic orbit) as the
time t tends to infinity. Then, it is shown that the stability boundary of this class
of nonlinear systems consists of the union of the stable manifolds of all equilibrium
states on the stability boundary (Chiang et al. 1988). With all these results, a numer-
ical procedure can be set up to determine the stability boundary by means of the
construction of the stable manifolds of all the equilibria which belong to it.
As an example, after computing the two stable equilibrium points, (0, 0, 0) and
(−7.45, −7.45, −7.45), and the unstable equilibrium point (−2.45, −2.45, −2.45) of
the following system:
˙x = −x + y,
˙y = 0.1x + 2y − x2
− 0.1x3
,
˙z = −y + z,
the procedure enables us to compute the stability boundary of the attracting domain
of the stable equilibrium points as the stable manifold of the unstable equilibrium
point. A partial view of it is shown in figure 15.
For higher-dimensional systems, graphic representation of the boundary is diffi-
cult. Nevertheless, their projection in particular subspaces gives useful information.
Figure 16 shows a partial view of the stability boundary between two stable equi-
Phil. Trans. R. Soc. Lond. A (1998)

18. 2198 P. Guicheteau
Figure 16. Projection in the (roll rate, yaw rate) plane of the boundary of the domain of
stability. PES1 and PES2 are stable equilibrium points while PEI1 is an unstable one.
librium points of a set of five nonlinear equations describing aircraft motion at low
AOA under inertia-coupling conditions.
(e) Transient behaviour
In many studies, the value of the parameter is considered to be fixed and inde-
pendent of time. In many practical situations, this is not the case, and the system
exhibits quasi-stationary behaviour and transient motions; the difference between
these two behaviours is the value of parameter change over time. If it is slow in
comparison to changes of state variables, quasi-stationary behaviour is observable.
There are at least two situations to be considered:
(1) movement along a stable branch;
(2) movement through bifurcation points.
In the first situation, it has been shown that if the system has a stable manifold
and fixed points corresponding to constant inputs, then an initial state close to this
manifold and a slowly varying input signal, in an average sense, produce a trajectory
that remains close to the manifold.
The second situation leads to different behaviour regarding the nature of the bifur-
cation point encountered. As an illustration, figure 17 shows possible situations.
Case (a) corresponds to a simple bifurcation point. A solution continues after the
bifurcation point along a stable branch. In case (b), where the bifurcation is also a
limit point, the branch on which the solution continues is chosen at random. In prac-
tical applications, the behaviour of the system has to be formulated statistically; the
character of distribution of fluctuations of state variables determines the probability
of the choice of individual branches of solutions. Case (c) is a very interesting one
because after crossing the limit point the system evolves into the closest stable state,
i.e. a state in whose domain of attraction the limit point belongs. The last case, (d),
corresponds to the Hopf bifurcation. Generally, the apparition of the stable periodic
orbit seems to be delayed and the low-amplitude solution around the bifurcation
point is unobservable (Neishtadt 1987).
Phil. Trans. R. Soc. Lond. A (1998)

19. Bifurcation theory 2199
Figure 17. Examples of possible crossing bifurcation and limit points during
quasi-stationary behaviour.
7. Conclusion
The behaviour of a fighter aircraft at AOA flight is so complex that it is very diffi-
cult to predict it exhaustively. Usually, this flight domain is investigated by means of
systematic or Monte Carlo numerical simulations before the first flight and by means
of extensive and expensive flight tests. Thanks to bifurcation theory and computer
capabilities, a methodology and software have been set up to investigate asymp-
totic behaviour of nonlinear differential equations depending on parameters. This
methodology is mainly used at ONERA to study the high-AOA behaviour of very
realistic missile, aircraft and submarine models. It is also used in fields other than
flight dynamics.
Coming back to the Alpha-Jet application, it can be seen that bifurcation theory
has been used to identify an aerodynamic model suitable for the analysis of high-
AOA flight regimes. To validate the methodology, flight tests were performed after
prediction of aircraft behaviour by means of equilibrium surfaces and periodic-orbit
envelopes for several aerodynamic formulations. Thanks to flight-test pilots, who
were asked to perform rather unusual flight tests, very good correlation with results
predicted by the theory has been obtained.
Thus, considering these results, it can be said that this technique has great poten-
tial and is appropriate for the investigation of aircraft behaviour, using only wind-
tunnel data. However, one cannot forget that the quality of prediction is directly
related to the quality of the aerodynamic database of the aircraft model.
As expected, another issue of the flight tests was that asymptotic behaviour was
not always reached because of the duration of the tests. Furthermore, in many real
problems, controls are not fixed or independent of time. It may follow transient
motions and/or quasi-stationary behaviour which must be addressed. These problems
are connected to the determination of the attracting basin of a stable equilibrium.
Much work has been done in this field. It is mainly based on Liapounov’s stability
theory and its extensions by Zubov and La Salle. More recently, introducing topologi-
cal considerations, trajectory-reversing methods have been developed, and numerous
computational procedures have been proposed. These procedures are appropriate to
Phil. Trans. R. Soc. Lond. A (1998)

20. 2200 P. Guicheteau
low-dimensional dynamic systems. However, there is a need to improve them for
higher dimensions. Due to the difficulty of working with high-dimensional systems,
a limited part of the attracting basin is generally computed. Is it sufficient? Are we
interested in the entire domain of attraction? It seems that a lot of work is needed
to give practical answers to this problem.
As a further step, the methodology can also be used to investigate nonlinear
behaviour induced by nonlinear elements in flight-control systems. From a stability
point of view and under implicit assumptions on the controlled system (continu-
ous time, continuous nonlinearities, etc.) it has been shown that bifurcation theory
can help the designer predict system behaviour and to compute ‘good’ control laws.
Although promising results are available in the literature, some work remains in order
to take into account practical systems. In this field, it seems also very interesting to
complete the analysis by determining the region of asymptotic stability to quantify
control-law robustness.
Finally, much work has been done on continuous systems. In practical applications,
digital flight control and nonlinearities in control modify the behaviour of continuous
systems. So, it becomes necessary to be able to numerically predict the behaviour
of complex systems with a continuous part and a discrete-time part. This requires
theoretical developments and modifications of numerical techniques which are of
interest for ONERA.
Some of the results presented here have been obtained under STPA (Services Techniques des
Programmes A´eronautiques) contracts. This was done in connection with Dassault Aviation for
the aircraft, and with the Flight Test Centre in Istres and ONERA/IMFL for the realization
and analysis of flight tests.
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